The epsilon-delta definition of continuity has a natural analog for functions that take lattice points into lattice points. It turns out that a function f is ‘continuous’ if and only if it takes neighbors into neighbors, i.e., if Q is a neighbor of P, then \(f(Q)=f(P)\) or f(Q) is a neighbor of f(P) (diagonal neighbors not allowed). Some basic properties of such ‘continuous’ functions are established. In particular, we show that a ‘continuous’ function from a finite block of lattice points into itself has an ‘almost-fixed’ point P such that f(P) is a neighbor of P (diagonal neighbors allowed). We also show that a function is one-to-one and continuous if and only if it is a combination of translations, rotations by multiples of \(90^ o\), or reflections at a horizontal, vertical, or diagonal line.